On 12/17/2012 11:47 AM, Bruno Marchal wrote:
On 16 Dec 2012, at 20:28, meekerdb wrote:
On 12/16/2012 2:31 AM, Bruno Marchal wrote:
No. With the CTM the ultimate truth is arithmetical truth, and we cannot really define
it (with the CTM). We can approximate it in less obvious ontologies, like second order
logic, set theory, etc. But with CTM this does not really define it.
Don't confuse truth, and the words pointing to it. Truth is always beyond words, even
the ultimate 3p truth.
What would it mean to 'define truth'? We can define 'true' as a property of sentence
that indicates a fact.
That's the best definition of some useful local truth. But when doing metaphysics, you
have to replace facts by "facts in some model/reality".
OK. But then it's "True relative to the model." and it's not necessarily The
But I'm not sure how to conceive of defining mathematical 'true'.
It is the object of model theory. You always need to add more axiom in a theory to
handle its model. You cannot define the notion of truth-about-set in ZF, but you can
define truth-about-set in ZF in the theory ZF +kappa (existence of inaccessible cardinals).
PA can define all the notion of truth for the formula with a bounded restriction of the
So what is that definition?
Does it just mean consistent with a set of axioms,
No. That means only having a model. true in some reality. But for arithmetic "true"
means satisfied by the usual structure (N, +, *).
i.e. not provably false?
How is not provably false different from 'satisfied by the usual structure'? Can you give
That just consistent.
I would think it was incompleteness. Consistency means not being able to prove every
proposition. But in a consistent system there can be propositions that are neither
provable nor disprovable. Are those true?
True entails consistency, but consistency does not entail truth.
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